Everybody in a room shakes hands with everybody else. The total number of handshakes is $45$. The total number of persons in the room is

In how many ways can $6$ persons be selected from $4$ officers and $8$ constables such that at least one officer is included?

In a Mathematics examination,there are $20$ questions of equal marks. The question paper is divided into three sections: $A, B$,and $C$. $A$ student is required to attempt a total of $15$ questions,taking at least $4$ questions from each section. If section $A$ has $8$ questions,section $B$ has $6$ questions,and section $C$ has $6$ questions,then the total number of ways a student can select $15$ questions is:

If $\binom{n+1}{3} = 2 \times \binom{n}{2}$,then $n = \dots$

The number of ways in which $15$ identical gold coins can be distributed among $3$ persons such that each one gets at least $3$ gold coins is

If a person has $3$ coins of different denominations,the number of different sums that can be formed is